Задача 1.

n = 0: \( \sum_{i=0}^{0}F_i = 0 \)

n = 1: \( \sum_{i=0}^{1}F_i = F_0 + F_1 = 0 + 1 = 1 = F_{1+2} - 1 = F_3 - 1 = 2 - 1 = 1\)

переход: \( \sum_{i=0}^{n}F_i = \sum_{i=0}^{n-1}F_i + F_n = F_{n-1+2} - 1 + F_n = F_{n} + F_{n+1} - 1 = F_{n+2} - 1 \)


Задача 2.

n = 1: \( \sum_{i=0}^{1}F^2_i = F^2_0 + F^2_1 = 0^2 + 1^2 = 1 = F_1\cdot F_2 = 1\cdot1 = 1 \)

n = 2: \( \sum_{i=0}^{2}F^2_i = F^2_0 + F^2_1 + F^2_2 = 0^2 + 1^2 +1^2 = 2 = F_2\cdot F_3 = 1\cdot2 = 2 \)

переход: \( \sum_{i=0}^{n}F^2_i = \sum_{i=0}^{n-1}F^2_i + F^2_n = F_{n-1}\cdot F_n + F^2_n = F_n(F_{n-1} + F_n) = F_n\cdot F_{n+1} \)
